How can one formalize the fact that the law of $X+Z$ where $X \in \mathbb{R}^d$ is any vector-valued random variable and $Z\sim \mathcal{N}(0, \sigma^2 \mathbf{I}_d)$ closely resembles the law of $Z$ if $\sigma^2$ is sufficiently large ? $X$ and $Z$ are supposed independent. Ideally, I would like to prove that some distance/divergence between the law of $X+Z$ and $Z$ approaches zero as $\sigma^2\to \infty$.

$\begingroup$This was meant as a comment ... Not an "answer" for now, but let's see how the discussion goes. Thanks.$\endgroup$
– mjwMar 26 at 18:47

$\begingroup$$X$ is any fixed vector-valued random variable, i.e. the law of $X$ cannot change or depend on the law of $Z$.$\endgroup$
– NocturneMar 26 at 19:18

$\begingroup$$X$ is constant? or a random variable?$\endgroup$
– mjwMar 26 at 19:39

$\begingroup$Okay, $X\sim \mathcal{N}(0, \sigma_0^2 \mathbf{I}_d)$, for example. A fixed distribution, whereas you want to show what happens as the distribution of $Z$ spreads out, yes?$\endgroup$
– mjwMar 26 at 19:43